weak* convergence in Sobolev space

Question

Consider $\Omega \subset \mathbb{R}^{N}$, open and bounded. If $u_{n} \rightharpoonup^{*} u$ in $W^{1,\infty}_{0}(\Omega)$, then does it follow that $u_{n} \rightharpoonup^{*} u$ in $W^{1,\infty}(\Omega)$?

Thanks.

Answer 1

It looks to me that since $W_0^{1,\infty}\subset W^{1,\infty}$, we have $(W^{1,\infty})^*\subset (W_0^{1,\infty})^*$. Therefore, if $u_n\to u$ weak star in $W_0^{1,\infty}$, it automatically weak star with respect to $W^{1,\infty}$.